"Immune to" literally means that someone is protected from a disease due to innoculation. "The Salk vaccine made a generation of children immune to polio."
It can also be used figuratively; for example, certain ambassadors and dignitaries have "diplomatic immunity," meaning they cannot be prosecuted for certain crimes.
"Free from" is a temporary condition. Just because you were free from colds last year doesn't mean you won't catch one this year.
"The dentist said I didn't need any fillings - I managed to stay free from tooth decay since my last visit."